If you are planning to go from Almaty to Shymkent over land by public transport, there are 2 ways: by train and by bus.
Both will take an equal amount of time, and cost roughly the same amount of money. The train is, however, much more comfortable than the bus.
Almaty – Shymkent by train
Many trains daily run from Almaty west towards Shymkent, and onwards all the way to Aktau. Get tickets for one of those in one of the many rail offices around town, from a Kazakh tour operator or on the train station of Almaty. The train to Shymkent takes 12 hours, an overnight train is ideal. Price is around 1500 tenge for platzkart. If you are traveling in summer, book in advance, trains fill up quickly.
Almaty – Shymkent by bus
Intercity buses to Shymkent leave from Sairan bus station in the western part of Almaty. A bus ticket will cost around 1300 tenge. The ride takes around 13 hours, buses go during the daytime as well as overnight.